The empty space around n disjoint line segments in the plane can be
partitioned into n + 1 convex faces by extending the segments
in some order. The dual graph of such a partition is the plane graph
whose vertices correspond to the n + 1 convex faces, and
every segment endpoint corresponds to an edge between the two incident faces
on opposite sides of the segment. We construct, for every set of n
disjoint line segments in the plane, a convex partition whose dual graph is
2-edge connected.